Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 489 https://internationalpubls.com On the Chromatic Restrained Domination Number of Strong Product of Graphs R. Divinelin Kumari 𝟏, M. K. Angel Jebitha πŸβˆ— and S. Sujitha 3 1 Research Scholar (Reg. No. 21213042092002), 2,3 Assistant Professor, 1,2,3 PG & Research Department of Mathematics, Holy Cross College (Autonomous), Nagercoil,Tamil Nadu, India. (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India) Email: 1divinelinr@gmail.com, 2angeljebitha@holycrossngl.edu.in, 3sujitha.s@holycrossngl.edu.in Article History: Received: 12-01-2024 Revised: 15-02-2024 Accepted: 01-03-2024 Abstract: Let 𝐺 = (𝑉, 𝐸) be a graph. A subset 𝐷 of 𝑉 is said to be a chromatic restrained dominating set (or crd-set) if 𝐷 is a restrained dominating set and πœ’(< 𝐷 >) = πœ’(𝐺). The minimum cardinality taken over all minimal chromatic restrained dominating sets is called the chromatic restrained domination number of 𝐺 and is denoted by π›Ύπ‘Ÿ 𝑐(𝐺). In this paper, we obtain the chromatic restrained domination number for the strong product of some standard graphs. Keywords : Domination, Restrained Domination, Chromatic Number, Strong Product. AMS Subject Classification : 05C15, 05C69 1. Introduction All the graphs 𝐺 = (𝑉, 𝐸) = (𝑛, π‘š) considered here are simple, finite and undirected, with neither loops nor multiple edges. For 𝐷 βŠ† 𝑉, the subgraph induced by 𝐷 is denoted by ⟨𝐷⟩. A k-vertex- coloring of a graph, or simply a k-coloring, is an assignment of k-colors to its vertices. The coloring is proper if no two adjacent vertices are assigned the same color. A coloring in which k-colors are used is a k-coloring. A graph is k-colorable if it has a proper k-coloring. The minimum π‘˜ for which a graph mailto:divinelinr@gmail.com mailto:angeljebitha@holycrossngl.edu.in mailto:sujitha.s@holycrossngl.edu.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 490 https://internationalpubls.com 𝐺 is k-colorable is called its chromatic number, and denoted by πœ’(𝐺). Graph Theory terminologies which are not defined here can be seen in [1] and [2]. A set 𝐷 βŠ† 𝑉 of vertices in a graph 𝐺 is called a dominating set if every vertex 𝑒 ∈ 𝑉 is either an element of 𝐷 or is adjacent to an element of 𝐷. The minimum cardinality taken over all minimal dominating sets is called the domination number of 𝐺 and is denoted by 𝛾(𝐺). A set 𝐷 βŠ† 𝑉 is a restrained dominating set if every vertex in 𝑉 βˆ’ 𝐷 is adjacent to a vertex in 𝐷 and another vertex in 𝑉 βˆ’ 𝐷 [3]. The minimum cardinality taken over all minimal restrained dominating sets is called the restrained domination number of 𝐺 and is denoted by π›Ύπ‘Ÿ(𝐺). A set 𝐷 is a π›Ύπ‘Ÿ - set if 𝐷 is a restrained dominating set of cardinality π›Ύπ‘Ÿ(𝐺). Strong product of two graphs 𝐺 and 𝐻 is the graph 𝐺 ⊠ 𝐻 whose vertex set is 𝑉(𝐺) Γ— 𝑉(𝐻), vertices (𝑒, π‘₯) and (𝑣, 𝑦) being adjacent if and only if 𝑒𝑣 ∈ 𝐸(𝐺) and π‘₯ = 𝑦 (or) 𝑒 = 𝑣 and π‘₯𝑦 ∈ 𝐸(𝐻) or 𝑒𝑣 ∈ 𝐸(𝐺) and π‘₯𝑦 ∈ 𝐸(𝐻) [4]. T. N. Janakiraman and M. Poobalaranjani introduced the concept of chromatic preserving set. A set 𝐷 βŠ† 𝑉 is a chromatic preserving set or a cp-set if πœ’(< 𝐷 >) = πœ’(𝐺) and the minimum cardinality taken over all cp-sets in 𝐺 is called the chromatic preserving number or cp-number of 𝐺 and is denoted by 𝑐𝑝𝑛(𝐺) [5]. A subset 𝐷 of 𝑉 is said to be a dom-chromatic set (or dc-set)if 𝐷 is a dominating set and πœ’(< 𝐷 >) = πœ’(𝐺). The minimum cardinality taken over all minimal dom- chromatic sets in 𝐺 is called the dom-chromatic number and is denoted by π›Ύπ‘β„Ž(𝐺)[6]. In this paper, the chromatic restrained domination number on the strong product of some standard graphs are obtained. 2 Main Results In this section, we obtain the chromatic restrained domination number for the strong product of some standard graphs. Definition 2.1 Let 𝐺 = (𝑉, 𝐸) be a graph. A subset 𝐷 of 𝑉 is said to be a chromatic restrained dominating set (or crd-set) if 𝐷 is a restrained dominating set and πœ’(< 𝐷 >) = πœ’(𝐺). The minimum cardinality taken over all minimal chromatic restrained dominating sets is called the chromatic restrained domination number and is denoted by π›Ύπ‘Ÿ 𝑐(𝐺). Throughout this paper, we denote the chromatic restrained domination number on the strong product of two graphs 𝐺 and 𝐻 by π›Ύπ‘Ÿ 𝑐(𝐺 ⊠ 𝐻). Theorem 2.2 For π‘Ÿ, 𝑠 β‰₯ 2, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾𝑠) = π‘Ÿπ‘ . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 491 https://internationalpubls.com Proof. Let 𝑉(πΎπ‘Ÿ) = {𝑒1, 𝑒2, 𝑒3, . . . , π‘’π‘Ÿ} and 𝑉(𝐾𝑠) = {𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑠}. Then, 𝑉(πΎπ‘Ÿ ⊠ 𝐾𝑠) = {(𝑒𝑖 , 𝑣𝑗)/1 ≀ 𝑖 ≀ π‘Ÿ, 1 ≀ 𝑗 ≀ 𝑠} where each vertex in πΎπ‘Ÿ ⊠ 𝐾𝑠 is a full degree vertex and |𝑉(πΎπ‘Ÿ ⊠ 𝐾𝑠)| = π‘Ÿπ‘ . Also, πœ’(πΎπ‘Ÿ ⊠ 𝐾𝑠) = π‘Ÿπ‘  as each vertex can be given different colors. Let 𝐷 be a π›Ύπ‘Ÿ 𝑐 βˆ’ set of πΎπ‘Ÿ ⊠ 𝐾𝑠. For any proper subset 𝑆 of 𝑉(πΎπ‘Ÿ ⊠ 𝐾𝑠), πœ’(βŸ¨π‘†βŸ©) < π‘Ÿπ‘ . Thus, 𝐷 = 𝑉(πΎπ‘Ÿ ⊠ 𝐾𝑠) is the only chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝐾𝑠. Therefore, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾𝑠) = |𝐷| = π‘Ÿπ‘ . Theorem 2.3 For 𝑠 β‰₯ 4, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Proof. Let 𝑉(πΎπ‘Ÿ) = {𝑒1, 𝑒2, 𝑒3, . . . , π‘’π‘Ÿ} and 𝑉(𝑃𝑠) = {𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑠}. Then 𝑉(πΎπ‘Ÿ ⊠ 𝑃𝑠) = {(𝑒𝑖 , 𝑣𝑗)/1 ≀ 𝑖 ≀ π‘Ÿ, 1 ≀ 𝑗 ≀ 𝑠} with cardinality π‘Ÿπ‘ . Also, πΎπ‘Ÿ ⊠ 𝑃𝑠 contains π‘Ÿ rows and 𝑠 columns 𝑉1, 𝑉2, 𝑉3, . . . , 𝑉𝑠. Since πœ’(πΎπ‘Ÿ) = π‘Ÿ and πœ’(𝑃𝑠) = 2, each column 𝑉𝑖, 𝑖 is odd can be colored with π‘Ÿ colors and each 𝑉𝑗, 𝑗 is even can be colored with another π‘Ÿ colors. Thus, πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠) = 2π‘Ÿ. Case (i): 𝑠 ≑ 0(π‘šπ‘œπ‘‘ 3) Let 𝐷 = {(𝑒1, 𝑣3π‘˜βˆ’1)/1 ≀ π‘˜ ≀ 𝑠 3 } where |𝐷| = 𝑠 3 . Then, 𝐷 is a dominating set and there does not exists any isolated vertex in βŸ¨π‘‰ βˆ’ 𝐷⟩. Thus, 𝐷 is a restrained dominating set and π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷| = 𝑠 3 . Since, 𝛾(𝑃𝑠) = 𝑠 3 and each vertex in column 𝑉𝑗 is adjacent to all the vertices in π‘‰π‘—βˆ’1, 𝑉𝑗 and 𝑉𝑗+1, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) β‰₯ 𝑠 3 . Thus, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) = 𝑠 3 . But, every minimum restrained dominating set is independent and so, πœ’(⟨𝐷⟩) = 1 β‰  πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠). Thus, 𝐷 is not a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠. Consider 𝐷1 = 𝐷 βˆͺ {(𝑒𝑖 , 𝑣2), (𝑒𝑗 , 𝑣3)/2 ≀ 𝑖 ≀ π‘Ÿ, 1 ≀ 𝑗 ≀ π‘Ÿ}. Since (𝑒1, 𝑣2) ∈ 𝐷, ⟨{(𝑒𝑖 , 𝑣2), (𝑒𝑗 , 𝑣3)/1 ≀ 𝑖, 𝑗 ≀ π‘Ÿ}⟩ is a complete subgraph on 2π‘Ÿ vertices and so, πœ’(⟨𝐷1⟩) = 2π‘Ÿ = πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠). Clearly, 𝐷1 is a restrained dominating set. Thus, 𝐷1 is a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠 and π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷1| = 𝑠 3 + 2π‘Ÿ βˆ’ 1 = π‘ βˆ’3 3 + 2π‘Ÿ = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Then, it remains to show that, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Since, πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠) = 2π‘Ÿ and πΎπ‘Ÿ ⊠ 𝑃𝑠 contains induced subgraph which is complete on 2π‘Ÿ vertices, any minimum chromatic restrained dominating set must contain those 2π‘Ÿ vertices which are the vertices of two adjacent columns. Let them be 𝑉2 and 𝑉3, so that, all the vertices of 𝑉1 and 𝑉4 are adjacent to the vertices of 𝑉2 and 𝑉3. From the remaining 𝑠 βˆ’ 4 columns, choose a vertex of each column 𝑉3(π‘˜+1), 1 ≀ π‘˜ ≀ ⌈ π‘ βˆ’4 3 βŒ‰ which is adjacent to all the vertices of 𝑉3(π‘˜+1) βˆ’ 1, 𝑉3(π‘˜+1) and 𝑉3(π‘˜+1) + 1. Thus, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Therefore, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 492 https://internationalpubls.com Case (ii): 𝑠 ≑ 1(π‘šπ‘œπ‘‘ 3) Let 𝐷2 = {(𝑒1, 𝑣3π‘˜βˆ’1)/1 ≀ π‘˜ ≀ ⌊ 𝑠 3 βŒ‹} βˆͺ {(𝑒1, 𝑣𝑠)} where |𝐷2| = ⌊ 𝑠 3 βŒ‹ + 1 = ⌈ 𝑠 3 βŒ‰. Then, every vertex of 𝑉 βˆ’ 𝐷2 is adjacent to at least one vertex of 𝐷2 and at least one another vertex in 𝑉 βˆ’ 𝐷2. Thus, 𝐷2 is a restrained dominating set and π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷2| = ⌈ 𝑠 3 βŒ‰. Since, 𝛾(𝑃𝑠) = ⌈ 𝑠 3 βŒ‰ and each vertex in column 𝑉𝑗 is adjacent to all the vertices in π‘‰π‘—βˆ’1, 𝑉𝑗 and 𝑉𝑗+1, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ 𝑠 3 βŒ‰. Therefore, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ 𝑠 3 βŒ‰. Consider 𝐷3 = 𝑉2 βˆͺ 𝑉3 βˆͺ {(𝑒1, 𝑣3(π‘˜+1))/1 ≀ π‘˜ ≀ π‘ βˆ’4 3 }. Clearly, 𝐷3 is a restrained dominating set and βŸ¨π‘‰2 βˆͺ 𝑉3⟩ is a complete graph on 2π‘Ÿ vertices. Thus, πœ’(⟨𝐷3⟩) = 2π‘Ÿ = πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠) and so, 𝐷3 is a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠. Then, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷3| = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Since πœ’(πΎπ‘Ÿ ⊠ 𝑃𝑠) = 2π‘Ÿ, any minimum chromatic restrained dominating set must contain all the π‘Ÿ vertices of two adjacent columns. Let them be 𝑉2 and 𝑉3 which is adjacent to all the vertices of 𝑉1 and 𝑉4. Again from the remaining 𝑠 βˆ’ 4 columns, choose a vertex from columns 𝑉3(π‘˜+1), 1 ≀ π‘˜ ≀ π‘ βˆ’4 3 . Thus, we get a minimum chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠 and π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Therefore, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Case (iii): 𝑠 ≑ 2(π‘šπ‘œπ‘‘ 3) Clearly, 𝐷2 is a restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠 and π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ 𝑠 3 βŒ‰. Also, 𝐷1 = 𝐷2 βˆͺ {(𝑒𝑖 , 𝑣2), (𝑒𝑗 , 𝑣3)/2 ≀ 𝑖 ≀ π‘Ÿ, 1 ≀ 𝑗 ≀ π‘Ÿ} is a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝑃𝑠 and so, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 2π‘Ÿ. Theorem 2.4 For π‘Ÿ, 𝑠 β‰₯ 2, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = 2π‘Ÿ. Proof. Let 𝑉(πΎπ‘Ÿ) = {𝑒1, 𝑒2, 𝑒3, . . . , π‘’π‘Ÿ} and 𝑉(𝐾1,𝑠) = {𝑣0, 𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑠} where 𝑣0 is the full degree vertex of 𝐾1,𝑠. Then, 𝑉(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = {(𝑒𝑖 , 𝑣𝑗)/1 ≀ 𝑖 ≀ π‘Ÿ, 0 ≀ 𝑗 ≀ 𝑠} and |𝑉(πΎπ‘Ÿ ⊠ 𝐾1,𝑠)| = (𝑠 + 1)π‘Ÿ. Clearly, πΎπ‘Ÿ ⊠ 𝐾1,𝑠 contains π‘Ÿ rows and 𝑠 + 1 columns (𝑉1, 𝑉2, 𝑉3, . . . , 𝑉𝑠+1) where the induced subgraph of πΎπ‘Ÿ ⊠ 𝐾1,𝑠 formed from all the vertices of two columns 𝑉1 and 𝑉𝑖 , 𝑖 β‰  1 is a complete subgraph on 2π‘Ÿ vertices. Then, πœ’(βŸ¨π‘‰1 βˆͺ 𝑉2⟩) = 2π‘Ÿ and all the remaining vertices can be colored using π‘Ÿ colors used for coloring the column 𝑉2, since there does not exists adjacency between columns 𝑉2, 𝑉3, . . . , 𝑉𝑠+1. Thus, πœ’(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = 2π‘Ÿ. Clearly, 𝐷 = {(𝑒1, 𝑣0)} is a restrained dominating set, as (𝑒1, 𝑣0) is a full degree vertex of πΎπ‘Ÿ ⊠ 𝐾1,𝑠. Therefore, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = 1. But, πœ’(⟨𝐷⟩) = 1 β‰  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 493 https://internationalpubls.com πœ’(πΎπ‘Ÿ ⊠ 𝐾1,𝑠), and so 𝐷 is not a chromatic restrained dominating set. Consider 𝐷1 = 𝑉1 βˆͺ 𝑉2, where πœ’(⟨𝐷1⟩) = 2π‘Ÿ = πœ’(πΎπ‘Ÿ ⊠ 𝐾1,𝑠). Since (𝑒1, 𝑣0) ∈ 𝐷1, 𝐷1 is also a restrained dominating set. Thus, 𝐷1 is a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ 𝐾1,𝑠 and π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) ≀ |𝐷1| = |𝑉1| + |𝑉2| = 2π‘Ÿ. Since πœ’(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = 2π‘Ÿ, any minimum chromatic restrained dominating set must contain a minimum of 2π‘Ÿ vertices. Therefore, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) β‰₯ 2π‘Ÿ. Hence, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ 𝐾1,𝑠) = 2π‘Ÿ. Theorem 2.5 For π‘Ÿ, 𝑠, π‘š β‰₯ 2, π›Ύπ‘Ÿ 𝑐(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = 2π‘š. Proof. Let 𝑉(πΎπ‘š) = {𝑒1, 𝑒2, 𝑒3, … , π‘’π‘š} and 𝑉(πΎπ‘Ÿ,𝑠) = {𝑣1, 𝑣2, 𝑣3, … , π‘£π‘Ÿ , π‘£π‘Ÿ+1, π‘£π‘Ÿ+2, … , π‘£π‘Ÿ+𝑠}. Then, 𝑉(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = {(𝑒𝑖 , 𝑣𝑗)/1 ≀ 𝑖 ≀ π‘š, 1 ≀ 𝑗 ≀ π‘Ÿ + 𝑠}. Clearly, πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠 contains π‘š rows and π‘Ÿ + 𝑠 columns where 𝑉1, 𝑉2, . . . , π‘‰π‘Ÿ+𝑠 denotes the columns. Also, the induced subgraph of πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠 formed from all the vertices of two columns, one among the columns 𝑉1, 𝑉2, . . . , π‘‰π‘Ÿ and another among the columns π‘‰π‘Ÿ+1, π‘‰π‘Ÿ+2, . . . , π‘‰π‘Ÿ+𝑠 is a complete subgraph on 2π‘š vertices. Clearly, the columns 𝑉1, 𝑉2, . . . , π‘‰π‘Ÿ can be colored with π‘š colors and the remaining columns π‘‰π‘Ÿ+1, π‘‰π‘Ÿ+2, . . . . , π‘‰π‘Ÿ+𝑠 can be colored with another π‘š colors. Thus, πœ’(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = 2π‘š. Consider a vertex from one of the columns 𝑉1, 𝑉2, . . . , π‘‰π‘Ÿ and another vertex from one of the columns π‘‰π‘Ÿ+1, π‘‰π‘Ÿ+2, . . . , π‘‰π‘Ÿ+𝑠. So, let 𝐷 = {(𝑒1, π‘£π‘Ÿ), (𝑒1, π‘£π‘Ÿ+1)}. Clearly, 𝐷 is a restrained dominating set and π›Ύπ‘Ÿ(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = 2. But, πœ’(⟨𝐷⟩) = 2 β‰  πœ’(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) and so, 𝐷 is not a chromatic restrained dominating set of πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠. Consider 𝐷1 = π‘‰π‘Ÿ βˆͺ π‘‰π‘Ÿ+1. Then, 𝐷1 is a restrained dominating set and πœ’(⟨𝐷1⟩) = |π‘‰π‘Ÿ| + |π‘‰π‘Ÿ+1| = 2π‘š = πœ’(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠). Thus, 𝐷1 is a chromatic restrained dominating set of πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠 and π›Ύπ‘Ÿ 𝑐(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) ≀ |𝐷1| = 2π‘š. Since πœ’(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = 2π‘š, any minimum chromatic restrained dominating set must contain at least 2π‘š vertices. Therefore, π›Ύπ‘Ÿ 𝑐(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) β‰₯ 2π‘š. Hence, π›Ύπ‘Ÿ 𝑐(πΎπ‘š ⊠ πΎπ‘Ÿ,𝑠) = 2π‘š. Theorem 2.6 For 𝑠 β‰₯ 5 and 𝑠 is odd, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = 3π‘Ÿ. Proof. Let 𝑉(πΎπ‘Ÿ) = {𝑒1, 𝑒2, 𝑒3, … , π‘’π‘Ÿ} and 𝑉(π‘Šπ‘ ) = {𝑣0, 𝑣1, 𝑣2, … , π‘£π‘ βˆ’1}. Then, 𝑉(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = {(𝑒𝑖 , 𝑣𝑗)/1 ≀ 𝑖 ≀ π‘Ÿ, 0 ≀ 𝑗 ≀ 𝑠 βˆ’ 1} where (𝑒1, 𝑣0) is the full degree vertex in πΎπ‘Ÿ ⊠ π‘Šπ‘ . Also, πΎπ‘Ÿ ⊠ π‘Šπ‘  consists of π‘Ÿ rows and 𝑠 columns, where 𝑉1, 𝑉2, … , 𝑉𝑠 denotes the columns. Clearly, 𝑉1 can be colored with π‘Ÿ colors, 𝑉2 can be colored with another π‘Ÿ colors , 𝑉3 can be colored with another π‘Ÿ colors and the remaining vertices can be colored with one among those 3π‘Ÿ colors. Then, πœ’(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = 3π‘Ÿ. Clearly, 𝐷 = {(𝑒1, 𝑣0)} is a restrained dominating set of πΎπ‘Ÿ ⊠ π‘Šπ‘ . Then, π›Ύπ‘Ÿ(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = 1. But, πœ’(⟨𝐷⟩) = 1 β‰  πœ’(πΎπ‘Ÿ ⊠ π‘Šπ‘ ). This implies that, 𝐷 is not a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ π‘Šπ‘ . Let 𝐷1 = 𝑉1 βˆͺ 𝑉2 βˆͺ 𝑉3 = {(𝑒1, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 494 https://internationalpubls.com 𝑣0), (𝑒2, 𝑣0), … , (π‘’π‘Ÿ, 𝑣0), (𝑒1, 𝑣1), (𝑒2, 𝑣1), . . . , (π‘’π‘Ÿ , 𝑣1), (𝑒1, 𝑣2), (𝑒2, 𝑣2), . . . , (π‘’π‘Ÿ , 𝑣2)}. Since the induced subgraph formed from all the vertices of 𝑉1, 𝑉2 and 𝑉3 is a complete graph on 3π‘Ÿ vertices, πœ’(⟨𝐷1⟩) = 3π‘Ÿ = πœ’(πΎπ‘Ÿ ⊠ π‘Šπ‘ ). Also, 𝐷1 is a restrained dominating set. Therefore, 𝐷1 is a chromatic restrained dominating set of πΎπ‘Ÿ ⊠ π‘Šπ‘  and π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) ≀ |𝐷1| = 3π‘Ÿ. Since πœ’(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = 3π‘Ÿ, any minimum chromatic restrained dominating set of πΎπ‘Ÿ ⊠ π‘Šπ‘  must contain a minimum of 3π‘Ÿ vertices and so π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) β‰₯ 3π‘Ÿ. Hence, π›Ύπ‘Ÿ 𝑐(πΎπ‘Ÿ ⊠ π‘Šπ‘ ) = 3π‘Ÿ. Theorem 2.7 For any π‘Ÿ, 𝑠 β‰₯ 2, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Proof. Let 𝑉(𝐾1,π‘Ÿ) = {𝑒0, 𝑒1, 𝑒2, 𝑒3, . . . , π‘’π‘Ÿ} where 𝑒0 is the full degree vertex of 𝐾1,π‘Ÿ and 𝑉(𝑃𝑠) = {𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑠}. Then, 𝑉(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = {(𝑒𝑖 , 𝑣𝑗)/0 ≀ 𝑖 ≀ π‘Ÿ, 1 ≀ 𝑗 ≀ 𝑠} and |𝑉(𝐾1,π‘Ÿ ⊠ 𝑃𝑠)| = (π‘Ÿ + 1)𝑠. Clearly, 𝐾1,π‘Ÿ ⊠ 𝑃𝑠 consists of π‘Ÿ + 1 rows and 𝑠 columns denoted as 𝑉1, 𝑉2, 𝑉3, . . . , 𝑉𝑠. Now, the first row can be colored with two colors and the remaining π‘Ÿ rows can be colored with extra two colors since no two vertices belonging to different rows (among those π‘Ÿ rows) are adjacent. This implies that, πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 4. Case (i): 𝑠 ≑ 0(π‘šπ‘œπ‘‘ 3) Let 𝐷1 = {(𝑒0, 𝑣3π‘˜βˆ’1)/1 ≀ π‘˜ ≀ 𝑠 3 } where |𝐷1| = 𝑠 3 . Then, 𝐷1 is a restrained dominating set since 𝐷1 is a dominating set and βŸ¨π‘‰ βˆ’ 𝐷1⟩ has no vertices of degree one. Thus, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷1| = 𝑠 3 . Since 𝛾(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 𝑠 3 , π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) β‰₯ 𝑠 3 . Therefore, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 𝑠 3 . Since 𝐷1 is an independent set, πœ’(⟨𝐷1⟩) = 1 β‰  πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) and so, 𝐷1 is not a chromatic restrained dominating set. Consider 𝐷2 = 𝐷1 βˆͺ {(𝑒0, 𝑣3), (𝑒1, 𝑣2), (𝑒1, 𝑣3)}. Then, ⟨𝐷2⟩ contains a complete subgraph on four vertices (𝑒0, 𝑣2), (𝑒0, 𝑣3), (𝑒1, 𝑣2) and (𝑒1, 𝑣3). Thus, πœ’(⟨𝐷2⟩) = 4 = πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠). Also, 𝐷2 is a restrained dominating set. This implies that, 𝐷2 is a chromatic restrained dominating set and π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷2| = |𝐷1| + 3 = 𝑠 3 + 3. Suppose there exists a chromatic restrained dominating set 𝑆1 such that |𝑆1| < 𝑠 3 + 3. Then |𝐷1| < |𝑆1| < 𝑠 3 + 3 = |𝐷1| + 3 and the only possible case for cardinality of 𝑆1 is either 𝑠 3 + 1 or 𝑠 3 + 2. But, there does not exists a chromatic restrained dominating set with cardinality 𝑠 3 + 1 or 𝑠 3 + 2. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 𝑠 3 + 3 = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Case (ii): 𝑠 ≑ 1(π‘šπ‘œπ‘‘ 3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 495 https://internationalpubls.com Let 𝐷3 = {(𝑒0, 𝑣3π‘˜βˆ’1)/1 ≀ π‘˜ ≀ π‘ βˆ’1 3 } βˆͺ {(𝑒0, 𝑣𝑠)} where |𝐷3| = π‘ βˆ’1 3 + 1 = ⌈ 𝑠 3 βŒ‰. Then, 𝐷3 is a dominating set and βŸ¨π‘‰ βˆ’ 𝐷3⟩ has no isolated vertices. Thus, 𝐷3 is a restrained dominating set and π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷3| = ⌈ 𝑠 3 βŒ‰. Since any minimum dominating set must contain the first vertex of each column 𝑉3π‘—βˆ’1, 1 ≀ 𝑗 ≀ π‘ βˆ’1 3 together with the first vertex of column 𝑉3 or the first vertex of each column 𝑉3𝑗+1, 1 ≀ 𝑗 ≀ π‘ βˆ’1 3 together with the first vertex of column 𝑉1 or the first vertex of each column 𝑉3𝑗, 1 ≀ 𝑗 ≀ π‘ βˆ’1 3 together with the first vertex of column 𝑉1 and so on, 𝛾(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = π‘ βˆ’1 3 + 1 = ⌈ 𝑠 3 βŒ‰. Thus, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ 𝑠 3 βŒ‰ and so, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = ⌈ 𝑠 3 βŒ‰. Since 𝐷3 is independent, πœ’(⟨𝐷3⟩) = 1 β‰  πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠). This implies that, 𝐷3 is not a chromatic restrained dominating set. Consider 𝐷4 = {(𝑒0, 𝑣2), (𝑒0, 𝑣3), (𝑒1, 𝑣2), (𝑒1, 𝑣3), (𝑒0, 𝑣3π‘˜)/2 ≀ π‘˜ ≀ π‘ βˆ’1 3 }. Then, 𝐷4 is a restrained dominating set since the columns 𝑉1, 𝑉2, 𝑉3 and 𝑉4 are dominated by the vertices (𝑒0, 𝑣2), (𝑒0, 𝑣3), (𝑒1, 𝑣2), (𝑒1, 𝑣3) and all the remaining vertices are adjacent to one of the vertex in {(𝑒0, 𝑣3π‘˜)/2 ≀ π‘˜ ≀ π‘ βˆ’1 3 }. Also, ⟨𝐷4⟩ contains 𝐾4 as an induced subgraph and so, πœ’(⟨𝐷4⟩) = 4. Therefore, 𝐷4 is a chromatic restrained dominating set of 𝐾1,π‘Ÿ ⊠ 𝑃𝑠 and π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷4| = π‘ βˆ’1 3 + 3 = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Suppose there exists a chromatic restrained dominating set 𝑆2 such that |𝑆2| < π‘ βˆ’1 3 + 3. Then |𝐷3| < |𝑆2| < π‘ βˆ’1 3 + 3 = |𝐷3| + 2 and the only possible cardinality of 𝑆2 is π‘ βˆ’1 3 + 2. But there does not exists a chromatic restrained dominating set with cardinality π‘ βˆ’1 3 + 2. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) β‰₯ π‘ βˆ’1 3 + 3. Hence, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = π‘ βˆ’1 3 + 3 = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Case (iii): 𝑠 ≑ 2(π‘šπ‘œπ‘‘ 3) Let 𝐷5 = {(𝑒0, 𝑣3π‘˜βˆ’1)/1 ≀ π‘˜ ≀ ⌈ 𝑠 3 βŒ‰} with cardinality ⌈ 𝑠 3 βŒ‰. Since 𝐷5 is a dominating set and βŸ¨π‘‰ βˆ’ 𝐷5⟩ has no isolated vertices, 𝐷5 is a restrained dominating set. Thus, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) ≀ |𝐷5| = ⌈ 𝑠 3 βŒ‰. Since 𝛾(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = ⌈ 𝑠 3 βŒ‰, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ 𝑠 3 βŒ‰. Therefore, π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = ⌈ 𝑠 3 βŒ‰. But every minimum restrained dominating set is independent and so, πœ’(⟨𝐷5⟩) = 1 β‰  πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠). This indicates that, 𝐷5 is not a chromatic restrained dominating set. Consider 𝐷6 = 𝐷5 βˆͺ {(𝑒0, 𝑣3), (𝑒1, 𝑣2), (𝑒1, 𝑣3)}. Then ⟨𝐷6⟩ contains 𝐾4 as an induced subgraph and so, πœ’(⟨𝐷6⟩) = 4 = πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠). Also, 𝐷6 is a restrained dominating set. Therefore, 𝐷6 is a chromatic restrained dominating set of 𝐾1,π‘Ÿ ⊠ 𝑃𝑠. Thus, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 496 https://internationalpubls.com 𝑃𝑠) ≀ |𝐷6| = ⌈ 𝑠 3 βŒ‰ + 3 = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Since πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 4 and any 𝐾1,π‘Ÿ ⊠ 𝑃𝑠 contains 𝐾4 as an induced subgraph, any chromatic restrained dominating set must contain all the four vertices of 𝐾4 so that, πœ’(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = 4. Let it be (𝑒0, 𝑣2), (𝑒0, 𝑣3), (𝑒1, 𝑣2), (𝑒1, 𝑣3) which dominates all the vertices in columns 𝑉1, 𝑉2, 𝑉3, 𝑉4. From the remaining 𝑠 βˆ’ 4 columns, choosing the first vertex of each column 𝑉3π‘˜+2, 1 ≀ π‘˜ ≀ ⌈ π‘ βˆ’4 3 βŒ‰, we get a minimum chromatic restrained dominating set. Thus, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) β‰₯ ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝑃𝑠) = ⌈ π‘ βˆ’4 3 βŒ‰ + 4. Theorem 2.8 Let π‘Ÿ, 𝑠 β‰₯ 2. Then π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = 4. Proof. Let 𝑉(𝐾1,π‘Ÿ) = {𝑒0, 𝑒1, 𝑒2, … , π‘’π‘Ÿ} and 𝑉(𝐾1,𝑠) = {𝑣0, 𝑣1, 𝑣2, … , 𝑣𝑠}. Then, 𝑉(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = {(𝑒𝑖 , 𝑣𝑗)/0 ≀ 𝑖 ≀ π‘Ÿ, 0 ≀ 𝑗 ≀ 𝑠} where (𝑒0, 𝑣0) is the full degree vertex of 𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠. Clearly, 𝐷 = {(𝑒0, 𝑣0)} is a restrained dominating set and π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = 1. Let 𝑉1, 𝑉2, 𝑉3, . . . , π‘‰π‘Ÿ+1 denotes the π‘Ÿ + 1 rows of 𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠 where 𝑉1 can be colored with two colors and the remaining π‘Ÿ rows can be colored with another two colors. Thus, πœ’(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = 4. But, πœ’(⟨𝐷⟩) = 1 β‰  πœ’(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) and so, 𝐷 is not a chromatic restrained dominating set. Let 𝐷1 = {(𝑒0, 𝑣0), (𝑒0, 𝑣1), (𝑒1, 𝑣0), (𝑒1, 𝑣1)} where ⟨𝐷1⟩ = 𝐾4. This implies that, πœ’(⟨𝐷1⟩) = 4 and 𝐷1 is also a restrained dominating set. Thus, 𝐷1 is a chromatic restrained dominating set of 𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠 and π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) ≀ |𝐷1| = 4. Since πœ’(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = 4, any minimum chromatic restrained dominating set must contain at least four vertices and so, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) β‰₯ 4. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ 𝐾1,𝑠) = 4. Theorem 2.9 For any π‘š, π‘Ÿ, 𝑠 β‰₯ 2, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = 4. Proof. Let 𝑉(𝐾1,π‘š) = {𝑒0, 𝑒1, 𝑒2, … , π‘’π‘š} and 𝑉(πΎπ‘Ÿ,𝑠) = {𝑣1, 𝑣2, 𝑣3, … , π‘£π‘Ÿ, π‘£π‘Ÿ+1, π‘£π‘Ÿ+2, … , π‘£π‘Ÿ+𝑠} where 𝑒0 is the full degree vertex of 𝐾1,π‘š. Now, 𝑉(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = {(𝑒𝑖 , 𝑣𝑗)/0 ≀ 𝑖 ≀ π‘š, 1 ≀ 𝑗 ≀ π‘Ÿ + 𝑠} and |𝑉(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠)| = (π‘š + 1)(π‘Ÿ + 𝑠). Also, there exists π‘š + 1 rows and π‘Ÿ + 𝑠 columns in 𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠. Clearly, the first row can be colored with two colors and the remaining π‘š rows can be colored with another two colors since there does not exists adjacency between any two vertices belonging to those π‘š different rows. Thus, πœ’(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = 4. Let 𝐷 = {(𝑒0, π‘£π‘Ÿ), (𝑒0, π‘£π‘Ÿ+1)}. Then 𝐷 is a restrained dominating set as 𝐷 is a dominating set and every vertex in 𝑉 βˆ’ 𝐷 is adjacent to at least one another vertex in 𝑉 βˆ’ 𝐷. Thus, π›Ύπ‘Ÿ(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) ≀ 2. Since there does not exists a full degree vertex in 𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠, π›Ύπ‘Ÿ(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) < 2 is impossible. Therefore, π›Ύπ‘Ÿ(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = 2. But πœ’(⟨𝐷⟩) = 2 and so, 𝐷 is not a chromatic restrained dominating set of 𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠. Consider 𝐷1 = 𝐷 βˆͺ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 497 https://internationalpubls.com {(𝑒1, π‘£π‘Ÿ), (𝑒1, π‘£π‘Ÿ+1)}. Then, ⟨𝐷1⟩ is a complete graph on four vertices and πœ’(⟨𝐷1⟩) = 4 = πœ’(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠). Also, 𝐷1 is a restrained dominating set. Thus, 𝐷1 is a chromatic restrained dominating set and π›Ύπ‘Ÿ 𝑐(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) ≀ |𝐷1| = 4. Since πœ’(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = 4, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) β‰₯ 4. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘š ⊠ πΎπ‘Ÿ,𝑠) = 4. Theorem 2.10 For any π‘Ÿ β‰₯ 3, 𝑠 β‰₯ 4, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = { 6 𝑖𝑓 𝑠 𝑖𝑠 π‘œπ‘‘π‘‘ 2𝑠 𝑖𝑓 𝑠 𝑖𝑠 𝑒𝑣𝑒𝑛 . Proof. Let 𝑉(𝐾1,π‘Ÿ) = {𝑒0, 𝑒1, 𝑒2, . . . , π‘’π‘Ÿ} and 𝑉(π‘Šπ‘ ) = {𝑣0, 𝑣1, 𝑣2, . . . , π‘£π‘ βˆ’1} where 𝑒0 and 𝑣0 are the full degree vertices of 𝐾1,π‘Ÿ and π‘Šπ‘  respectively. Then, 𝑉(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = {(𝑒𝑖 , 𝑣𝑗)/0 ≀ 𝑖 ≀ π‘Ÿ, 0 ≀ 𝑗 ≀ 𝑠 βˆ’ 1} where |𝑉(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ )| = (π‘Ÿ + 1)𝑠. Also, 𝑑𝑒𝑔(𝑒0, 𝑣0) = (π‘Ÿ + 1)𝑠 βˆ’ 1. Clearly, 𝐾1,π‘Ÿ ⊠ π‘Šπ‘  contains π‘Ÿ + 1 rows (𝑉1, 𝑉2, 𝑉3, . . . , π‘‰π‘Ÿ+1) and 𝑠 columns. Case (i): 𝑠 is odd Then the first row of 𝐾1,π‘Ÿ ⊠ π‘Šπ‘  can be colored with three colors and the second row can be colored with another three colors. Since there does not exists adjacency between any two vertices belonging to different rows of 𝑉2, 𝑉3, … , π‘‰π‘Ÿ+1, all the π‘Ÿ rows can be colored with three colors. Then, πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 6. Since (𝑒0, 𝑣0) is the full degree vertex, 𝐷 = {(𝑒0, 𝑣0)} is a restrained dominating set of 𝐾1,π‘Ÿ ⊠ π‘Šπ‘  and π›Ύπ‘Ÿ(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 1. But πœ’(⟨𝐷⟩) = 1 and so, 𝐷 is not a chromatic restrained dominating set. Let 𝐷1 = {(𝑒0, 𝑣0), (𝑒0, 𝑣1), (𝑒0, 𝑣2), (𝑒1, 𝑣0), (𝑒1, 𝑣1), (𝑒1, 𝑣2)} where ⟨𝐷1⟩ = 𝐾6 and |𝐷1| = 6. Then, πœ’(⟨𝐷1⟩) = 6 = πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) and 𝐷1 is a restrained dominating set. Therefore, 𝐷1 is a chromatic restrained dominating set and π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) ≀ |𝐷1| = 6. Since πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 6, any chromatic restrained dominating set must contain at least six vertices. Thus, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) β‰₯ 6. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 6. Case (ii): 𝑠 is even Then the first row 𝑉1 can be colored with four colors. Since, some of the vertices in 𝑉1 and 𝑉2 are adjacent, the second row 𝑉2 can be colored by introducing three more colors. Also, the remaining rows 𝑉3, 𝑉4, … , π‘‰π‘Ÿ+1 can be colored by assigning the same colors as in 𝑉2. Thus, πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 7. Let 𝐷2 = { (𝑒0, 𝑣0), (𝑒0, 𝑣1), (𝑒0, 𝑣2), … , (𝑒0, π‘£π‘ βˆ’1), (𝑒1, 𝑣0), (𝑒1, 𝑣1), (𝑒1, 𝑣2), . . . , (𝑒1, π‘£π‘ βˆ’2)}. Then πœ’(⟨𝐷2⟩) = 7 = πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ). But 𝐷2 is not a restrained dominating set since (𝑒1, π‘£π‘ βˆ’1) ∈ 𝑉 βˆ’ 𝐷2 has no adjacent vertex in 𝑉 βˆ’ 𝐷2. So, consider 𝐷3 = 𝐷2 βˆͺ {(𝑒1, π‘£π‘ βˆ’1)}. Then, πœ’(⟨𝐷3⟩) = 7 and 𝐷3 is a restrained dominating set. Thus, 𝐷3 is a chromatic restrained dominating set of 𝐾1,π‘Ÿ ⊠ π‘Šπ‘  and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 498 https://internationalpubls.com π›Ύπ‘Ÿ 𝑐𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) ≀ |𝐷3| = |𝐷2| + 1 = 2𝑠. Since 𝑠 is even and πœ’(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 7, any minimum chromatic restrained dominating set of 𝐾1,π‘Ÿ ⊠ π‘Šπ‘ must contain at least 2𝑠 vertices and so, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) β‰₯ 2𝑠. Therefore, π›Ύπ‘Ÿ 𝑐(𝐾1,π‘Ÿ ⊠ π‘Šπ‘ ) = 2𝑠. 3. Conclusion In this article, the chromatic restrained domination number on the strong product of certain standard graphs are obtained. A promising avenue for future research is to investigate the bounds on the strong product of graphs and identify the extremal graphs that define the upper and lower limits of the chromatic restrained domination number in such products. References [1] Frank Harary, Graph Theory, Addison - Wesley Publishing Company, 1969. [2] Teresa W. Haynes, Stephen T. Hedetniemi, Peter J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. [3] Bondy. J. A and Murty. U. S. R, Graph Theory with Applications, Springer, 2008. [4] Gayla S. Domke, Johannes H. Hattingh, Stephen T. Hedetniemi, Renu C. Laskar, Lisa R. Markus, Restrained Domination in Graphs, Discrete Mathematics, 203 (1999) 61 - 69. [5] Janakiraman. T. N and Poobalaranjani. M, On The Chromatic Preserving Sets, International Journal of Engineering Science, Advanced Computing and Bio - Technology, Vol. 1, No. 1, January - March 2010, pp. 29 - 42. [6] Janakiraman. T. N and Poobalaranjani. M, Dom-Chromatic sets of graphs, International Journal of Engineering Science, Advanced Computing and Bio - Technology, Vol. 2, No. 2, April - June 2011, pp. 88 - 103.